On a Control Design to an Afm Micro Cantilever Beam, Operating In

1 Copyright © 2011 by ASME

Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA

DETC2011-MNS 47543

ON A CONTROL DESIGN TO AN AFM MICROCANTILEVER BEAM, OPERATING IN A TAPPING-MODE, WITH IRREGULAR BEHAVIOR

Kleber dos Santos Rodrigues UNESP: Univ Estadual Paulista, Department of Mechanical Engineering, FEB, UNESP, Bauru,

São Paulo, Brazil

José Manoel Balthazar UNESP: Univ Estadual Paulista

, Department of Statistics, Applied Mathematics and Computation, DEMAC, Rio Claro, São Paulo,

Brazil

Angelo Marcelo Tusset Department of Engineering Science, UTFPR

Ponta Grossa, Paraná, Brazil

Bento Rodrigues Pontes Júnior UNESP: Univ Estadual Paulista,Department of

Mechanical Engineering, FEB, UNESP, Bauru, São Paulo, Brazil

ABSTRACT

In last decades, control of nonlinear dynamic systems became an important and interesting problem studied by many authors, what results the appearance of lots of works about this subject in the scientific literature. In this paper, an Atomic Force Microscope micro cantilever operating in tapping mode was modeled, and its behavior was studied using bifurcation diagrams, phase portraits, time history, Poincare maps and Lyapunov exponents. Chaos was detected in an interval of time; those phenomena undermine the achievement of accurate images by the sample surface. In the mathematical model, periodic and chaotic motion was obtained by changing parameters. To control the chaotic behavior of the system were implemented two control techniques. The SDRE control (State Dependent Riccati Equation) and Time-delayed feedback control. Simulation results show the feasibility of the both methods, for chaos control of an AFM system. INTRODUCTION Atomic Force Microscope (AFM) is a powerful tool in scanning probe microscope, its application includes manipulation of carbon nanotubes, DNA, imaging and actuation in nano-electronics, etc. (Rützel et al., 2003). In AFM, a micro-cantilever with a tip at its free end vibrates and sends a signal to a photo detector, the acquired images come from this movement and the scanning process starts.

Figure 1 – A Schematic of tapping mode atomic force microscope (Zhao and Danckowitz, 2006) The micro-cantilever has three modes of operation: non-contact, contact and tapping mode operation. Tapping mode is the most common type of operation, where the contact between tip and sample occur. To generate the images, the micro-cantilever vibrates near from its resonance. By this fact, the micro cantilever may exhibit chaotic behavior under contains circumstances (Yau et al, 2009).According to the literature, Sebastian et al. (2001) used a one-degree-of-freedom model, with linear coefficients to describe attractive and repulsive forces. Garcia and San Paulo (2000) used the spring-mass equation to describe the micro-cantilever behavior and DMT, and van der Waals theories to describe tip-sample interactions. After years, Misra and Danckowits (2007) proposed a model and used event-driven method to control the resonant behavior. Chaotic behavior is common in AFM. When this type of behavior occurs, the images are affected, what is not a desirable outcome. Mathematical modeling is commonly used to

Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA

DETC2011-47543


understand the behavior of AFM micro-cantilever. Numerical simulation and analysis of the obtained images are essential to project control methods and to stabilize the system. This paper utilizes the mathematical model proposed by Rützel et al (2003) which consists of using Bernoulli equations to describe the micro-cantilever dynamics behavior and Lennard Jones potential to describe the tip-sample interactions. In modeling process, a partial-differential equation occurs and is discretized and nondimensionalized via Galerking method. By using bifurcation diagrams, phase portrait, time history, Lyapunov exponents and Poincare maps, was possible to observe the chaotic and periodic behavior. Chaos is observed in a limited interval of time, what hardly interferes on the analysis of obtained images. Nonlinear systems can be identified as weakly non-linear and strongly non-linear. In AFM, weakly nonlinearities appear in non-contact mode and traditional perturbation methods like the average method and multiple scale method can be used (Nozaki et al, 2010; Aime´ et al. 1999; Boisgard et al. 1999; Nony et al. 2001; Couturier et al. 2002). Time delayed feedback control method was originally proposed by Pyragas for stabilizing unstable periodic orbits embedded in chaotic attractors. A key feature of the control method is that control input depends on only the difference signal between the current and past outputs of a nonlinear system. The control method, therefore, can be implemented without any identification of the model, system parameters, and underlying dynamics from experimental data (Yamasue and Hikihara, 2006), (Salarieh and Alasty, 2007). Tapping mode operation is identified as a strong nonlinear system (Rützel, et al), then the use of those perturbation methods become a complex job. First, because analytical solutions for nonlinear system are unknown in most cases, and second, the perturbation methods become more difficult to implement (Guran, 1997). To stabilize the system, the State-Dependent Riccati Equation (SDRE) control was used to make the synchronization between the nonlinear systems, driven the chaotic system (slave system), to the periodic system (master system). At least, feedfoward control was used to maintain the system controlled in periodic motion. The paper is organized as follows: modeling of micro cantilever deflection and the tip-sample interactions; discretization of the system’s equation; numerical simulations to find chaotic and periodic behavior; synchronization and stabilization of the system using SDRE control method; consider the stability of time delayed feedback control.

USED NOMENCLATURE

L micro cantilever length ρ mass density E Young’s modulus I area moment of inertia A cross section

*η equilibrium gap

z distance between tip and sample w*(L) deflection of micro cantilever tip

Ω piezoelectric frequency ( )q x piecewise manner

∆ Dirac’s delta R Probe tip radius

( )n xφ eingenfunction approximation

y micro cantilever tip displacement Y vibration amplitude nondimensionalized x state vector A dependent state matrix F nonlinear vector U control term uf feedfoward control u feedback control e error between the orbits (master and slave)

B two dimensional constant vector concerning coupling between the control input and the state variables.

P(x) Riccati state dependent solution Q, R LQR matrices C11, C12, d1, E1 non-dimensional parameters acquired by the Si-Si system under consideration. η non-dimensional amplitude of dither piezoelectric actuator.

1ρ , 2ρ number densities of molecules

1c , 2c interaction coefficients of intermolecular

pair Potentials

MATHEMATICAL MODEL

The Euler-Lagrange equation is used to describe the micro cantilever dynamics, and Lennard-Jones potential is used to represents the interactions between tip and sample. L is the cantilever length, ρ is mass density, E is the Young’s modulus, and area moment of inertia I of the cross-section area A is chosen in the analysis. The beam is clamped at x=0 and free at x=L. The micro cantilever deflection is u(x,t*),

* *( )Z w Lη = − is the equilibrium gap between probe tip and

sample, and the piezoelectric is modeled by ( *)y t Ysen t= Ω

(Rützel et al, 2003).


Figure 2 – A schematic of the (a) initial, (b) intermediate, and (c) current configurations associated with the micro cantilever deformation (Qin-Quan, Chen, 2006) MODELING OF THE CANTILEVER DEFLECTION .

In this case is assumed that cantilever behavior is similar with the beam behavior, than Euler–Bernoulli equation is used to describe the relationship between the beam’s deflection and the applied load (Witmer, 1991-1992):

4 4

4 4

( , *) *( )( )

u x t w xEI EI q x

x x

∂ ∂+ = ∂ ∂ (1)

The dynamic beam equation is the Euler-Lagrange beam equation:

2 22 2 2

2 20

1 ( , *) 1 ( , *) * ( )( ) ( , *)

2 * 2

L u x t u x t xS A EI q x u x t dx

t x x

ωρ ∂ ∂ ∂ = − + + ∂ ∂ ∂

∫

(2)

The first term represents the kinetic energy, the second one represents the potential energy due to internal forces and the third term represents the potential energy due to the external load. The Euler-Lagrange equation is used to determine the function that minimizes the functional S. For a dynamic Euler-Bernoulli beam, the Euler-Lagrange equation is

2 2 2 2

2 2 2 2

( , *) *( ) ( , *)( )

*

u x t x u x tEI A q x

x x x t

ω ρ ∂ ∂ ∂ ∂+ = − + ∂ ∂ ∂ ∂

(3)

Simplifying (3):

2 4 4

2 4 4

( , *) ( , *) * ( )( )

*

u x t u x t xA EI q x

t x x

ωρ ∂ ∂ ∂+ + = ∂ ∂ ∂

(4)

Where, q(x) represents the distributed load over the micro-cantilever lenght.

Boundary conditions:

Equation 4 is a fourth order partial differential equation and boundary conditions are needed to find solution( , )u x t . In this

case, the load ( )q x is represented in a piecewise manner, load

isn’t a continuous function. Representation of point load as a distribution using the Dirac function in this model results:

( )

2 4 4

2 4 4

( , *) ( , *) *( )

*

( )

d u x t d u x t d xA EI

dt dx dx

q x x L

ωρ

δ

+ + =

−

With:

0( , ) 0

xu x t

== ,

0

( , *)0

x

du x t

dx =

= , 2

2

( , *)0

x L

d u x t

dx=

=

The partial differential equation is:

( )( , *) ( '''' * ''''( )) ( )Au x t EI u x q x x Lρ ω δ+ + = −&&

(5)

In the next section, ( )q x is modeled as attraction/repulsion

force derived from Lennard-Jones interaction potential (Salarieh and Alasty, 2007).

MODELING OF THE TIP SURFACE INTERACTIONS .

Lennard-Jones does not model the real contact mechanics encountered in tapping mode AFM, but it represents a generic tip-surface-interaction potential (Ruetzel, Lee, Raman, 2003). According to Basso et al (2000) and Israelachvili (1991), the representation of interaction forces between the probe tip of radius R and the sample surface, with a gap z is:

1 2, 7 61260L J

A R A RU

zz= − , 1 2

, 8 2( )

180 6L J

L J

U A R A RP z

z z z

∂= − = + −

∂

Where A1 and A2 are the Hamaker constants, and represents repulsive and attractive potentials respectively. The Hamaker constants are given by 2

1 1 2 1A cπ ρ ρ= and, 22 1 2 2A cπ ρ ρ= .The

gap z is represented with * ( , *) sin( *)z u L t Y tη= − − Ω , than the

Lennard-Jones potential is:

1 2, 8 2180( * ( , *) sin( *)) 6( * ( , *) sin( *))L J

A R A RP

u L t Y t u L t Y tη η= − +

− − Ω − − Ω (6)


Finally, using equation 6 in equation 5, and substitution of q(x) by PL,J, is possible to obtain the governing equation of the system:

( , *) ( ( , *) * ( ))Au x t EI u x t xρ ω′′′′ ′′′′+ + =&&

1 2

8 2.

180( * ( , *) sin *) 6( * ( , *) sin *)

A R A R

u L t Y t u L t Y tη η

= − + − − Ω − − Ω

2. ( ) sin *x L A Y tδ ρ− + Ω Ω (7)

DISCRETIZATION

Next consider the situation when the excitation frequency Ω is close to the lowest frequency of the micro-cantilever. Under near-resonant forcing, and in the absence of additional internal resonances, only one mode of the micro cantilever is assumed to participate in the response (Qin-Quan and Chen, 2007):

1 1( , *) ( ) ( *)u x t x q tφ= (8)

Where 1( )xφ is the first approximate eigenfunction.

Substitution of (8) into (7), multiplication of (7) by 1( )xφ ,

subsequent integration over the domain, and the introduction of a modal damping consistent with the Q factors listed in table table1 (appendix 2) yields the single-degree-of-freedom model:

211 12

1 1 18 2(1 ) (1 )

C Cy d y y B E sin t

y sin t y sin tη

η η= − − + + + + Ω Ω

− − Ω − − Ω&& &

(9)

A positive y is the micro cantilever tip displacement towards the sample, nondimensionalized by the equilibrium gap between the tip and the sample,11c , 12c , 1d and 1E , are non-dimensional

parameters acquired by the Si-Si system under consideration. In this case, η is the vibration amplitude of the dither piezoelectric actuator nondimensionalized by the equilibrium gap width. Considering the following substitutions: yx =1 , yx &=2 . The

differential equation (9) can also be represented in state space:

)sin())sin(1(

))sin(1(1

28

1

611211

12112

21

tEtx

txCCBxdxx

xx

ΩΩ+Ω−−

Ω−−+++−−=

=

ηη

η&

&

(10) SYSTEM WITH CHAOTIC BEHAVIOR Numerical simulations with (10) using software Matlab, integrator ODE45, step length h=0.001, parameters;

1 0.01d = , 1 0.148967B = − , 511 4.59118 10C −= − × , 1 1.57367E =

12 0.149013C = , with initial conditions (0) 0.2x = and (0) 0x =&

follows:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

x

n

Figure 3 – Bifurcation diagram In figure 3, it’s possible to observe that with η=0.9 the first bifurcation is observed. With η=0.9, the following simulations are obtained: In figures 4, 5 and 6, with 900 1000t≤ ≤ , chaotic

motion appears in the system, with 51.2 10t = × , Lyapunov exponents obtained the results:

1 2 30.010154, 0.020146, 0λ λ λ= = − = .With one of the

exponent is positive, the system is chaotic for 510t ≤ .

900 910 920 930 940 950 960 970 980 990 1000-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t

x 1

Figure 4 – Time history of chaotic motion.


Figure 5 – Poincare map

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-4

-3

-2

-1

0

1

2

3

4

5

x1

x 2

Figure 6 – Phase portrait of chaotic motion

0 2 4 6 8 10 12

x 104

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

Lyap

unov

Exp

onen

ts

Figure 7 – Positive Lyapunov exponents

SYSTEM WITH PERIODIC MOTION

Numerical simulations carried with Eq.(10), using software

Matlab, integrator ODE45, steplenght h=0.001, and parameters;

1 0.01d = , 0.01η = , 1 0.148967B = − , 511 4.59118 10C −= − × ,

12 0.149013C = , 1 50E = , we will obtain through figures 8 and

9, with 900 1000t≤ ≤ , periodic motion. :

900 910 920 930 940 950 960 970 980 990 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

x 1

Figure 8 – Time history of periodic motion

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5

x1

x 2

Figure 9 – phase portrait of periodic motion

SYNCHRONIZATION OF NONLINEAR SYSTEMS

In oscillators, synchronization can occurs spontaneously, be induced or even forced by control techniques. Two or more oscillators can be taken to artificially synchronize their trajectories; such technique is justified in practical applications, when the synchronization is necessary. In synchronization, the desired trajectory of a slave system is the trajectory of the master system, and the control signal (feedback) at a given moment is the function of difference between the trajectories an instant before. In equation 10, with parameters of periodic motion being the master system, and chaotic motion as the slave system, the objective is to take the chaotic motion of slave


system to periodic motion of the master system. Next, figure 9 shows both systems in same phase portrait:

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-4

-3

-2

-1

0

1

2

3

4

5

x1

x 2

periodic

chaotic

Figure 10 – Phase portrait of periodic motion and chaotic motion

STATE DEPENDENT RICATTI EQUATION (SDRE) AND FEEDFOWARD CONTROL

Since the mid-90’s, State-Dependent Riccati Equation (SDRE) strategies have emerged as general design methods that provide a systematic and effective means of designing nonlinear controllers, observers, and filters. One of the reasons to choose SDRE control method is the fact that many others control techniques developed have very limited applicability because of the strong conditions imposed on the system (Tayfun, 2008). In this paper, two control principles are used : the feedfoward to maintain the slave system in the desired periodic orbit of the master system, and the state feedback control that is used to stabilize the behavior of dynamic systems. Consider the nonlinear system (10) in this form: ( )x A x x F= +& (11)

Where nx R∈ is the state vector, n nA R ×∈ is a dependent state matrix, F is the nonlinear vector of terms not state dependent. With control, the system (11) is:

( )x A x x F U= + +& (12)

With:

fU u u= + (13)

fu is the feedfoward control:

fu F= − (14)

With 0(0)x x= , and substitution of (13) in (12), the system

(12) is represented in matrix form:

( )x A x x Bu= +& (15)

Where n nB R ×∈ is matrix control, and u is the linear state feedback control. Find u using State-Dependent Riccati Equation Method. Writing the dynamic system defined by (10):

1 2

2 1 1 2

x x

x x d x F

== − − +

&

& (16)

Where: 6

211 12 11 18

1

(1 sin( ))sin( )

(1 sin( ))

C C x tF B E t

x t

ηη

η+ − − Ω

= + + Ω Ω− − Ω

(17) By introducing (13) in (16), follows (Shawky et al., 2007):

ux

x

dx

x

+

−−=

1

0

1

10

2

1

12

1

&

& (18)

The system in (17), implies that the origin of the system is an equilibrium point, condition that allows using the SDRE control to calculate u. Considering the system (18) as (15) the matrix A, and matrix B, are represented by:

−−=

11

10

dA ,

0

1B

=

(19)

The feedback control u, is:

1( ) ( ) ( ) ( )Tu R x B x P x K x e−= − = − (20)

with: *( )e x x= − , x is the slave system signal and *x is the

master system signal, P(x) is the Riccati state dependent solution:

1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0T TP x A x A x P x P x B x R x B x P x Q x−+ − + =

(21) By using:

=

3

3

100

010Q , [ ]-510=R

The functional to be minimized by the SDRE control is given by:

0

( ( ) ( ) )T Tr rJ e Q e e u R u u dt

∞

= +∫ (22)

The synchronization of these two systems can be observed in figure 11, the error *( )e x x= − in figure 10 and the phase

portrait in figure 11, after application of control (13) in (10).


0 10 20 30 40 50 60 70 80 90 100-2

-1.5

-1

-0.5

0

0.5

1

t

x 1

master

slave

(a)

6.5012 6.5013 6.5014 6.5015 6.5016 6.5017 6.5018 6.5019 6.502-1.25

-1.25

-1.2499

-1.2498

-1.2498

-1.2498

-1.2497

-1.2496

-1.2496

-1.2496

-1.2495

t

x 1

master

slave

(b)

Figure 11 – a: Synchronization of master and slave systems, b: zoom of the figure (a)

-2 -1.5 -1 -0.5 0 0.5 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x 2

master

slave

(a)

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.20.0384

0.0385

0.0386

0.0387

0.0388

0.0389

x1

x 2

master

slave

(b)

Figure 12 – (a): Phase portrait of two systems, master and slave controlled, (b): zoom of the figure (a)

We observed from figures 10 and 11 that control application, was efficient, driven the slave system to the master system, with

0003.0max * =− xx for 100≤t .

TIME DELAYED FEEDBACK CONTROL

So far, this control method has been successfully applied to various experimental systems including the AMF (Yamasue and Hikihara, 2006). As originally suggested by Pyragas, continuous control input u stabilizing a chaotic oscillation is given by the difference between the current output and the past one as follows (Yamasue and Hikihara, 2006), (Pyragas, 1992, 2002): ( ) ( )[ ] ( ) ( )[ ] txtxgtxtxgKu 2121 ,, −−−= ττ (23)

where: τ is the time delay and k the feedback gain. The ( ) ( )[ ]txtxg 21 , and ( ) ( )[ ]ττ −− txtx 21 , imply a scalar output

signal measured at the current time t and the past time( )τ−t ,

respectively. Since the control input (1) only depends on the output signal. The time delay τ is adjusted to the period of a target unstable periodic orbit we intend to stabilize in a chaotic attractor, and the control input therefore converges to null after the controlled system is stabilized to the target orbit (Yamasue and Hikihara, 2006).Assuming that the velocity of oscillation is measured as an output of the nonlinear system (10), the control signal u is given as follows: ( ) ( )[ ]txtxKu 22 −−= τ (24)


The system (10), with control (24) m can be represented in the

following way:

( ) ( )[ ]txtxKtE

tx

txCCBxdxx

xx

2212

81

611211

12112

21

)sin(

))sin(1(

))sin(1(

−−+ΩΩ

+Ω−−

Ω−−+++−−=

=

τηη

η&

&

(25)

The time delay τ and feedback gain K are important control parameters which substantially affecting of the control

performance. The time delay τ is adjusted to ππτ 22 =Ω

= to

stabilize an orbit with the same frequency as the external force oscillating the microcantilever, and the feedback gain is here adjusted to be 2.0=K , as proposed in Yamasue and Hikihara (2006). Figures 13 and 14 shows the application of control (24).

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-5

-4

-3

-2

-1

0

1

2

3

4

5

x1

x 2

without control

control

(a): t=[1:600]

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-5

-4

-3

-2

-1

0

1

2

3

4

5

x1

x 2

without control

control

(b): t=[10:600]

Figure 13 - Chaotic attractor (black) and target unstable periodic orbit (red)

0 10 20 30 40 50 60 70 80 90 100-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t

x 1

without control

control

Figure 14 - Displacement of microcantilever

Considering the above results, we can see that time-delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. CONCLUSIONS

Control and stabilization of chaotic and irregular behavior of micro-cantilevers in AFM are essential, because when this kind of phenomena occurs, there’s a lost of quality in image detection process. Modeling of a control method in tapping-mode is very difficult due to strong nonlinearities. In this case, is possible to observe in figures 9, 10 and 11 that the method of synchronization, with the proposed control method was efficient with easy implementation. SDRE was effective, the implementation was very simple, and the synchronization used, can be implemented in practical situations without problems, with satisfactory results. The numerical simulation shows that error can be minimized to acceptable values by adjusting matrices Q and R of LQR control, with a global stability of controlled system. In this paper with the given Q and R matrices, the obtained

synchronization has an error 0003.0max * =− xx , after the

synchronization, feedfoward control kept the slave system into master system condition, which ensured the efficiency of methods, used to control the system. In figures 13 and 14 we can observe the feasibility in applying of the time delayed feedback control technique for both chaos control of AFM system (10). Analyzing of the application of the two controls, we can verify that The Possibility of application using time delayed feedback depends on dynamical properties of target periodic orbits under grazing bifurcation, and the system takes time to synchronize with the π2=t desired orbit, while the application of the SDRE control enables its application to stabilize any desired orbits, with much less time needed to π2=t as observed in the time delayed feedback control.


ACKNOWLEDGMENTS

Authors thank CNPq and Fapesp for financial support. K.S. Rodrigues thanks CAPES for fellowship support.

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Brazilian conference on dynamics, control and their Applications (http://www.sbmac.org.br/dincon/ ) Nozaki, R., 2010, “Nonlinear dynamics, chaos and control in Atomic Force Microscope”, |Master Degree dissertation, mech Enginering College (FEB), UNESP, SP, Brazil. Pyragas, K., 1992, “Continuous control of chaos by self-controlling feedback”, Phys. Lett. A, 170, pp.421-428. Pyragas, K., 2002, “Analytical properties and optimization of time-delayed feedback control”, Physical Review E 66, 026207, pp. 1-9. Qin-Quan H. and Chen L., 2007,“Bifurcation and chaos in atomic force microscope.” Elsevier, Chaos, Solutions and potentials”, Proc R Soc London A ; 459:1925–48. Rutzel, S., Lee, S.I. and Raman, A., 2003, “Nonlinear dynamics of atomic-force-microscope probes driven in Lenard-Jones”, Fractals Vol 33, pp. 711–715. Salarieh, H, A. Alasty, 2007, “Control of chaos in atomic force microscopes using delayed feedback based on entropy minimization”, Communications in Nonlinear Science and Numerical Simulation Vol 14, pp. 637- 644 Sebastian, A., Salapaka, M. V., Chen, D. J., and Cleveland J. P., 2001, “Harmonic and power balance tools for tapping mode atomic force microscope”, Journal of Applied Physics Vol 89, pp. 6473–6480. Shawky, A. M., Ordys, A. W., Petropoulakis, L. and Grimble, M. J., 2007, “Position control of flexible manipulator using non-linear with state-dependent Riccati equation”. Proc. IMechE, 221 Part I: J. Systems and Control Engineering, pp. 475- 486. Witmer, E.A., 1992. "Elementary Bernoulli-Euler Beam Theory". MIT Unified Engineering Course Notes. pp. 114-164 Yamasue, K. and Hikihara, T., 2006, “Control of microcantilevers in dynamic force microscopy using time delayed feedback”, Review of Scientific Instruments 77, 053703, pp. 1-6. Yau , H., Wang C., Chou, Y., 2009. “Design of Sliding Mode Controller for AFM system by Backstepping Design” Proc. IEEE International Conference on Networking, Sensing and Control, pp. 840-845


ANNEX A

DISCRETIZATION PROCESS

Consider the partial differential equation;

1 2

8 2

2

( , *) ( ( , *) * ( ))

.180( * ( , *) sin *) 6( * ( , *) sin *)

. ( ) sin *.

Au x t EI u x t x

A R A R

u L t Y t u L t Y t

x L A Y t

ρ ω

η η

δ ρ

′′′′ ′′′′+ + =

= − + − − Ω − − Ω

− + Ω Ω

&&

Now, consider the situation when the excitation frequency Ω is close to the lowest frequency of the micro-cantilever. Under near-resonant forcing, and in the absence of additional internal resonances, only one mode of the micro cantilever is assumed to participate in the response (Qin-Quan and Chen, 2007):

1 1( , ) ( ) ( )u x t x q tφ=

Where 1( )xφ is the first approximate eigenfunction.

Substitution of (8) into (7), multiplication of (7) by 1( )xφ ,

subsequent integration over the domain, and the introduction of a modal damping consistent with the Q factors listed in table 1, annex B yields the single-degree-of-freedom model (Rützel, Lee and Raman, 2003):

211 121 1 18 2(1 ) (1 )

C Cy d y y B E sin t

y sin t y sin tη

η η= − − + + + + Ω Ω

− − Ω − − Ω&& &

Where

1( *)

*

x ty

η= , 1 1 1( *) ( ) ( *)x t L q tφ= , * *( )Z w Lη = − ,

0*t tω= , 1 1(1 )B α= − Γ , *

Yηη

= ,3

3EIk

L= ,

0ωΩΩ = ,

11

20 10

L

cd

A dxω ρ φ=

∫,

*

Zαη

= , 111 19180 ( *)

A RC

k η= − Γ ,

212 136 ( *)

A RC

k η= − Γ ,

1 11 00

210

L

L

EI dx

A dx

φ φω

ρ φ

′′′′= ∫

∫,

21

12 20 10

( )L

k L

A dx

φ

ω ρ φΓ =

∫

1 101

210

( )L

L

L dxE

dx

φ φ

φ= ∫

∫

and a positive y s the microcantilever tip displacement towards the sample, non-dimensionalized by the equilibrium gap between the tip and the sample. 1η is the vibration

amplitude of the dither piezoelectric actuator non-dimensionalized by the equilibrium gap width, and 0ω is a

reference frequency for subsequent non-dimensionalization. Note that B1 + C1l + C12 = 0, so that, on the substitution of y = u = 0 in (8), equation (8) is identically satisfied. The dotted quantities now represent derivatives with respect to rescaled time t.

ANNEX B

TABLE 1 - PROPERTIES AND DIMENSIONS OF THE CANTILEVERS (RÜTZEL, LEE AND RAMAN, 2006). Descriptions symbol Si–Si(111) Si–polystyrene length L 449 µm 154 µm width b 46 µm 13.7 µm thickness h 1.7 µm 6.9 µm tip radius R 150 nm 20 nm material density ρ 2 330 kg m−3 2 330 kg m−3 static stiffness k 0.11 N m−1 40 N m−1 elastic modulus E 176 GPa 130GPa 1st resonance f1 11.804 kHz 350.0 kHz Q factor (air) Q 100 100 Hamaker (att.) A2 1.865×10−19J 1.150×10−19J Hamaker (rep.) A1 1.3596×10-70Jm6 0.838×10-70J

On a Control Design to an Afm Micro Cantilever Beam, Operating In

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